Fields: basic examples

Here is the simplest example, called the Boolean field: $F=\{0,1\}$, with two binary operations, $+$ (addition mod 2), and $\cdot$ (integer multiplication). This field is used is the mathematics of electrical circuits ($1$ being ``on'' and 0 being ``off'').

In general, a field is a set $F$ which has two binary operations, denoted $+$ and $\cdot$, satisfying the usual algebraic properties which the reals ${\mathbb{R}}$ or complexes ${\mathbb{C}}$ satisfy.

Definition 2.1.1   A field is a set $F$ which has two binary operations, denoted $+$ and $\cdot$, satisfying the following properties. For all $a,b,c\in F$, we have

1. $a+b=b+a$, (``addition is commutative'')
2. $a\cdot b=b\cdot a$, (``multiplication is commutative'')
3. $(a+b)+c=a+(b+c)$, (``addition is associative'')
4. $(a\cdot b)c=a(b\cdot c)$, (``multiplication is associative'')
5. $(a+b)\cdot c=a\cdot c+b\cdot c$, (``distributive'')
6. there is an element $1\in F$ such that $a\cdot 1=a$, (``$1$ is a multiplicative identity'')
7. there is an element $0\in F$ such that $a+0=a$ (``0 is a additive identity''),
8. if $a\not= 0$ then there is an element, denoted $a^{-1}$, such that $a\cdot a^{-1}=1$ (``the inverse of any non-zero element exists'').

Example 2.1.2   We have seen several examples of fields already: $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{F}_p =\mathbb{Z}/p\mathbb{Z}$, where $p$ is a prime.

A word of advertisement. Since the set $\mathbb{F}_p$ is finite, we call it (at least when $p$ is a prime number) a finite field. This field is often also denoted $GF(p)$. There are other finite fields besides $\mathbb{F}_p$. For each prime power $q=p^k$ there is a finite field, usually denoted $\mathbb{F}_q$ or $GF(q)$. In later subsections, we shall introduce methods of constructing these fields. A word of caution: $\mathbb{Z}/n\mathbb{Z}$ is never a field unless $n$ is a prime (the proof of this is left as an exercise).

If $F$ satisfies (1)-(7) but not necessarily (8) then $F$ is called a ring ^2.22.3. The subset of elements of a ring $F$ whose inverse do exist is denoted by $F^\times$. If $F$ satisfies (1),(3)-(8) but not necessarily (2) then $F$ is called a skew field.

If there is an integer $n>0$ such that , for all $x\in F$, $n\cdot x =0$ then the smallest such integer is called the characteristic of $F$. If there is no such integer then we say that $F$ is characteristic 0.

Example 2.1.3   The fields $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$ all have characteristic 0.

If $E$ and $F$ are fields and

(a) as sets, $F\subset E$,

(b) the field operations for $F$ are the restrictions of the field operations for $E$,

then $F$ is called a subfield of $E$ and $E$ is called an extension field of $F$, written $E/F$.

In particular, every field must contain at least two elements (namely, 0 and $1$). The field containing exactly two elements is called the Boolean field and denoted ${\mathbb{F}}_2$ (or $GF(2)$). It is of characteristic $2$.

Example 2.1.4  

(1) $\mathbb{Q}$ is a field. $\mathbb{R}$ is a field. $\mathbb{C}$ is a field.

(2) If $p$ is a prime then ${\mathbb{F}}_p$ is a field.

(3) ${\mathbb{Z}}/n{\mathbb{Z}}$ is a ring. (It is not a field unless $n$ is a prime.) For example, $\overline{2}\not= \overline{0}$ in ${\mathbb{Z}}/4{\mathbb{Z}}$ but $\overline{2}^{-1}$ does not exist. (If $\overline{2}^{-1}$ did exist, then since $\overline{2}\cdot \overline{2}=\overline{0}$, by the cancellation law, $\overline{2}= \overline{2}\cdot \overline{2}\cdot \overline{2}^{-1}= \overline{0}\cdot \overline{2}^{-1}=\overline{0}$, a contradiction.)

Lemma 2.1.5   If $F$ is a field with characteristic $n>0$ then $n$ is a prime number.

proof: If not, $n=ab$ and $nx=abx=0$ for all $x$, where $a<n$ and $b<n$. If $ax\not= 0$ for some $x\not= 0$ in $F$ then (by the cancellation property) $ax\not= 0$ for all $x$. Conversely, if $ax= 0$ for some $x\not= 0$ in $F$ then (by the cancellation property) $ax= 0$ for all $x$. These two facts allow us to conclude that either $ax= 0$ for all $x$ in $F$ or $bx= 0$ for all $x$ in $F$. Since $n$ was defined to be the smallest such integer, this is contradiction. $\Box$

A field $F$ and let $p(x)=a_0+a_1x+...+a_{n-1}x^{n-1}+x^n$ be a polynomial having coefficients in $F$ (so $a_i\in F$). Some fields, such as $=\mathbb{C}$ have the property that all the roots of $p(x)=0$ belong to $\mathbb{C}$. This fact is called the fundamental theorem of algebra.

Theorem 2.1.6   fundamental theorem of algebra Let $F=\mathbb{C}$. Let $p(x)=a_0+a_1x+...+a_{n-1}x^{n-1}+x^n$ be any polynomial with coefficients in $F$ (so $a_i\in F$). Then $p(x)=0$ has exactly $n$ roots in $\mathbb{C}$ (counted with multiplicity).

Some fields do not have the property that every polynomial has all its roots in the field. For example, if $F=\mathbb{Q}$ and $p(x)=x^2+1$ then $p(x)=0$ has no roots in $F$ (though it has, by the above theorem, $2$ roots in $\mathbb{C}$).

Definition 2.1.7   A field $F$ satisfying the property that every polynomial of degree $n$ with coefficients in $F$ has all its coefficients in $F$ is called algebraically closed.

For example, $\mathbb{C}$ is algebraically closed but $\mathbb{Q}$ is not. Though we shall not prove it here, every field $F$ is contained in an algebraically closed field $K$. If $K$ is an algebraically closed field containing $F$ and there is no algebraically closed subfield of $K$ containing $F$ then we call $K$ an algebraic closure of $F$, written $K=\overline{F}$.

If $F$ is not algebraically closed then it is possible to construct extension fields of $F$. These constructions are explored in the next several sections.